Optimal regularity for the fully nonlinear thin obstacle problem

Abstract

In this work, we establish the optimal regularity for solutions to the fully nonlinear thin obstacle problem. In particular, we show the existence of an optimal exponent ${\alpha }_F$ such that $u$ is $C^{1,{\alpha }_F}$ on either side of the obstacle. In order to do that, we prove the uniqueness of blow-ups at regular points, as well as an expansion for the solution there. Finally, we also prove that if the operator is rotationally invariant, then ${\alpha }_F\ge \frac{1}{2}$ and the solution is always $C^{1,1/2}$.